Off-diagonal bounds for the Dirichlet–to–Neumann operator on Lipschitz domains

Abstract

Let Ω be a bounded domain of Rn+1 with n≥1. We assume that the boundary Γ of Ω is Lipschitz. Consider the Dirichlet–to–Neumann operator N0 associated with a system in divergence form of size m with real symmetric and Hölder continuous coefficients. We prove Lp(Γ)→Lq(Γ) off-diagonal bounds of the form‖1Fe−tN01Ef‖q≲(t∧1)nq−np(1+d(E,F)t)−1‖1Ef‖p for all measurable subsets E and F of Γ. If Γ is C1+κ for some κ>0 and m=1, we obtain a sharp estimate in the sense that (1+d(E,F)t)−1 can be replaced by (1+d(E,F)t)−(1+np−nq). Such bounds are also valid for complex time. For n=1, we apply our off-diagonal bounds to prove that the Dirichlet–to–Neumann operator associated with a system generates an analytic semigroup on Lp(Γ) for all p∈(1,∞). In addition, the corresponding evolution problem has Lq(Lp)-maximal regularity.